Intro
TROLL model currently compute leaf lifespan with Reich’s allometry (Reich et al. 1991). But we have shown that Reich’s allometry is underestimating leaf lifespan for low LMA species. Moreover simulations estimated unrealistically low aboveground biomass for low LMA species. We assumed Reich’s allometry underestimation of leaf lifespan for low LMA species being the source of unrealistically low aboveground biomass inside TROLL simulations. We decided to find a better allometry with Wright et al. (2004) GLOPNET dataset.
Figure 1: Leaf mass per area (LMA), leaf nitrogen content (Nmass) and leaflifespan (LL). Leaf mass per area (LMA in \(g.m^{-2}\)), leaf nitrogen content (Nmass, in \(mg.g^-1\)) and leaf lifespan (LL in \(months\)) are taken in GLOPNET dataset from Wright et al. (2004).
Model
We tested different models and we kept subsequant model with the best tradeoff between log likelihood, number of parameters, and convergence : \[ {LL_s}_j \sim \mathcal{logN}(log({\mu_s}_j),\,\sigma)\,\] \[s=1,...,S_{=4}~, ~~j=1,...,n_s, ~~X^{(1)}=LMA, ~~X^{(2)}=Nmass, ~~i=0,...,I_{=2}\]
\[{\mu_s}_j = {\beta_0}_s*e^{{\beta_1}_s*{X_s}_j^{(1)} - {\beta_2}_s*{X_s}_j^{(2)}}\] \[{\beta_i}_s \sim \mathcal{N}({\beta_i},\,\sigma_i)\,^I\] \[(\beta_i, \sigma, \sigma_i) \sim \mathcal{\Gamma}(0.001,\,0.001)\,^{2I+1}\] Leaf lifespan (\(LL\)) is following a lognormal law with location \(log({\mu_s}_j)\) and scale \(\sigma\). \(s\) represents the dataset used to do the fit between 1 and \(S=4\), it encompass environmental and protocol variations. \(j\) represents the observation for a given site \(s\). Finally \(X^{(1)}\) represents the LMA and \(X^{(2)}\) the leaf nitrogen content (Nmass). Location \({\mu_s}_j\) is the log of the exponential from the difference between LMA and Nmass with parameters \({\beta_0}_s\), \({\beta_1}_s\), and \({\beta_2}_s\). Each \({\beta_i}_s\) is following a normal law located on \(\beta_i\) with scale \(\sigma_i\). Finally all \(beta_i\), \(\sigma\), and \(\sigma_i\) are assumed without presemption following gamma laws of parameters \(10^-2\), \(10^-2\).
Results
We obtained convergence for the model with a maximum likekihood of 10.0385366. Figure 2 shows posterior distributions and their associated markov chains. Figure 3 presents model predictions.
Mcomp
\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s} + {\beta_2}_s*N^{{\beta_4}_s},\sigma)\,\]
M1
\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA + {\beta_2}_s*N,\sigma)\,\] Maximum likekihood of 10.0385366
M2
\[ LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s} + N^{{\beta_4}_s},\sigma)\,\] Maximum likekihood of -1.731814
M3
\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s} + {\beta_2}_s*N^{{\beta_4}_s},\sigma)\,\] Maximum likekihood of 18.3651541
M4
\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA + N,\sigma)\,\] Maximum likekihood of 0.880405
M5
\[ LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s} + N^{{\beta_4}_s},\sigma)\,\] Maximum likekihood of -0.1829464
M6
\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s} + N^{{\beta_4}_s},\sigma)\,\] Maximum likekihood of 4.2540211
M7
\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA,\sigma)\,\] Maximum likekihood of 3.95695
M8
\[ LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s},\sigma)\,\] Maximum likekihood of -7.9790043
M9
\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s},\sigma)\,\] Maximum likekihood of 9.4630726
Figure 2: Model posterior distributions and associated markov chains.
Graph
If we keep model 9: \(LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s},\sigma)\,\)
Figure 3: Model predictions. \[LL = 1.529*e^{0.149 *LMA^{0.558}}\]
References
Reich, P.B., Uhl, C., Walters, M.B. & Ellsworth, D.S. (1991). Leaf lifespan as a determinant of leaf structure and function among 23 amazonian tree species. Oecologia, 86, 16–24.
Wright, I.J., Reich, P.B., Westoby, M., Ackerly, D.D., Baruch, Z., Bongers, F., Cavender-Bares, J., Chapin, T., Cornelissen, J.H.C., Diemer, M. & Others. (2004). The worldwide leaf economics spectrum. Nature, 428, 821–827.